Theory and Applications of Categories, Vol. 30, No. 43, 2015, pp. 1429–1468.
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO
2-GROUP ACTIONS
JEFFREY C. MORTON AND ROGER PICKEN
Abstract. Transformation groupoids associated to group actions capture the interplay between global and local symmetries of structures described in set-theoretic
terms. This paper examines the analogous situation for structures described in categorytheoretic terms, where symmetry is expressed as the action of a 2-group G (equivalently,
a categorical group) on a category C. It describes the construction of a transformation
groupoid in diagrammatic terms, and considers this construction internal to Cat, the
category of categories. The result is a double category C//G which describes the local
symmetries of C. We define this and describe some of its structure, with the adjoint
action of G on itself as a guiding example.
1. Introduction
Symmetry is a fundamental notion both in mathematics and in physics. In this paper,
we consider the symmetries of categories: in particular, if a symmetry of a category C
is represented as an endofunctor of C, one also has ’symmetries of symmetries’, because
there are natural transformations between these functors. These two levels of symmetry
can be represented by using a 2-group. Thus we consider actions of 2-groups on categories,
and develop the analog of the “transformation groupoid” construction, which captures the
local view of the symmetries for a group action on a set.
1.1. Symmetries, group actions and 2-group actions. The usual mathematical
representation of symmetry is as a group action on a set, or perhaps on a manifold, vector
space, or whatever sort of structure is of interest. Not only the objects on which the
groups act, but also the groups themselves may have various sorts of structure - one may
be interested in algebraic groups, or Lie groups, for example. In particular, symmetry
in this sense is described by the action of a group object in an appropriate category: an
object in the category equipped with structure maps, such as the multiplication map
m : G × G → G, satisfying the axioms which define a group.
The authors would like to acknowledge Dany Majard for many helpful discussions in the writing
of this paper. This work was supported in part by the Graduiertenkolleg 1670 “Mathematics Inspired
by String Theory and QFT” of the Mathematics Department of the University of Hamburg, and by
the Fundação para a Ciência e a Tecnologia, Portugal, through projects PTDC/MAT/101503/2008,
EXCL/MAT-GEO/0222/2012 and PEst-OE/EEI/LA0009/2013.
Received by the editors 2014-10-03 and, in revised form, 2015-10-07.
Transmitted by James Stasheff. Published on 2015-10-26.
2010 Mathematics Subject Classification: 18B40, 18D10, 20L99.
Key words and phrases: 2-group, categorical group, crossed module, action, double category, adjoint
action.
c Jeffrey C. Morton and Roger Picken, 2015. Permission to copy for private use granted.
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JEFFREY C. MORTON AND ROGER PICKEN
Even superficially very different constructions, such as quantum groups, fit this pattern
to some extent. A quantum group is a rather broad concept with many variations, but
the starting point is the idea of a Hopf algebra, a bialgebra with a certain structure.
Quantum groups play a role in describing symmetry in noncommutative geometry and
other algebraic settings. Hopf algebras may be seen as group objects in Algop , the opposite
category of some category of algebras. (Which category of algebras depends on the context
within noncommutative geometry).
The notion of a group action, also in the broader sense above, describes what are known
as global symmetries: global operations on the object which supports the symmetry,
leaving it in some sense unchanged. The exact sense in which it is to be unchanged,
and what a transformation of the object can consist of, is the guiding criterion in finding
its group of symmetries. On the other hand, there is also a local concept of symmetry.
Following Weinstein [28], local symmetry may be described in terms of groupoids. Recall
that groupoids are categories in which all morphisms are invertible. Now the entire set
which supports the symmetry, such as the set of points of a space, forms the objects of the
groupoid, and the morphisms of the groupoid represent symmetry relations which relate
one point to another, making them “indistinguishable” under the relation in question.
Such symmetry relations may or may not arise from a global group action on the set,
and thus the local “groupoid approach” to symmetry is more fine-grained than the global
“group action approach”.
Of course, given a global group action, it can also be viewed from an equivalent local
perspective, in terms of the transformation groupoid associated to the global group action.
Note that not every groupoid representing local symmetry is a transformation groupoid1 .
In our generalization of group actions to 2-group actions, we will develop both global and
local perspectives, and show them to be equivalent.
The construction of a transformation groupoid, coming from a group action on a
set, makes sense not only in the category of sets, but also in other contexts. Typical
examples would include a Lie group acting on a manifold, a topological group acting on
a topological space, or an algebraic group acting on a variety. In reasonable situations
(which always apply in a topos, such as the category of sets, or of topological spaces), one
can construct a transformation groupoid which captures the local picture implied by such
global symmetry groups. These will be, respectively, Lie groupoids, topological groupoids,
or algebraic groupoids in the cases mentioned above. We can say that each of these
situations is an instance of a general process, which can be described diagrammatically,
and independently of the type of objects playing the role of “sets”. In other words, there
is a general construction of a transformation groupoid, which makes sense in a category
(of a suitable kind). Each of the cases we mentioned are examples of this construction
1
Some easy counter-examples appear as sub-groupoids of transformation groupoids. For instance,
suppose we have an action of a Lie group G on a manifold M . We can consider a small neighborhood
U in M , and take the sub-groupoid of the transformation groupoid associated to the action of G with
only those objects corresponding to points in U . This has all the morphisms taking points in U to other
points in U , even if U does not contain any entire orbit of the G action. That is, it retains all the ’local’
information about M ’s symmetries that can be detected in U .
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1431
“internal to” a chosen ambient category, such as topological spaces, manifolds, varieties,
and so forth.
The situation we describe in this paper generalizes these ideas in a different direction.
We are interested in describing symmetry of categories by taking the notion of a group
action on a set to a higher categorical level, both for the group and for the set it acts on.
Thus we will be studying the action of a 2-group on a category.
In order to understand this step, it is useful to recall that a group is a particular type
of category, with just a single object and all morphisms invertible. The product on the
group is the composition of morphisms. Now, given an object X in a category, X, there
is a sub-category End(X) of X with one object X and morphisms consisting of all the
endomorphisms of X. Restricting to the invertible such morphisms, one gets Aut(X).
This is a subcategory of X, but it is also a group in the sense just mentioned. It is,
indeed, the full symmetry group of the object X. In these terms, the action of a group G
on X is described by a functor from the category G to the category of automorphisms of
X, say φ : G → Aut(X).
Now, just as a group may be regarded as a special sort of category, so too a 2-group can
be seen as a special kind of 2-category, namely one with just a single object and invertible
1- and 2-morphisms. Given a category C, i.e. an object in the 2-category Cat, one has
endofunctors from C to itself, but also natural transformations between such endofunctors.
So End(C) is a 2-category with a single object C, whose 1-morphisms are endofunctors
of C and whose 2-morphisms are natural transformations between endofunctors. If we
restrict to only those endofunctors and natural transformations which are invertible, we
have Aut(C), which is a 2-group. It is, indeed, the full symmetry 2-group of C. Then,
in complete analogy with the action of a group G on a set X, the action of a 2-group G
on a category C is described by a 2-functor from the 2-category G to the 2-category of
automorphisms of C:
Φ : G → Aut(C).
Intuitively, following Baez and Lauda in [1], we are considering not only symmetries of C
(the 1-morphism level of Φ), but also “symmetries between symmetries” (the 2-morphism
level of Φ).
As an aside, we note that there are other approaches to generalizing the action of
G on X to higher settings. Some recent work has been done on categories equipped
with actions of a group G, (note, not a 2-group) and their “equivariantizations” (see, for
example, [15, 24, 9] among others). These equivariantizations generalize the notion of the
fixed points of a group action on a (possibly structured) set. From our perspective this is
a rather special case of the most general sense of symmetry for categories since we expect
to have not only morphisms between categories (namely, functors), but also morphisms
between these morphisms (namely, natural transformations). This means that, for us,
the notion of “symmetry” of a category is inherently somewhat more subtle than that
for a set, or even for any set with an additional structure, such as a topological space.
Another generalization, going beyond ours, is to consider the action of 2-groupoids, or yet
more general categorical structures, on a category [6]. A 2-groupoid is a 2-category with
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JEFFREY C. MORTON AND ROGER PICKEN
invertible 1- and 2-morphisms, but it may have more than one object. However, we wish to
focus on the 2-group case, because it naturally generalizes the important and ubiquitous
notion of actions of groups, and because the application we have in mind (in higher gauge
theory, to be described shortly) motivated us to study 2-group actions, and not the action
of anything more general. Finally, we note the recent work by J. Elgueta [13], where a
weak 2-group acts on a groupoid by equivalences, rather than by endofunctors. Here the
self-equivalences of an object in a 2-category naturally constitute a weak 2-group. This
is not directly comparable to our approach, since weak 2-groups are used throughout in
Elgueta’s paper.
Returning to our main discussion, the essential point in this paper is that 2-group
actions, too, can be understood in terms of an internal construction like those described
above, i.e. transformation groupoids internal to some category. The ambient category
is now Cat, the category whose objects are (small) categories and whose morphisms are
functors. Then we can use the fact that an alternative way to see 2-groups is as categorical
groups, namely group objects in Cat. Starting with an ordinary group action on a set,
the corresponding transformation groupoid is obtained from a function, φ̂ : G × X → X,
satisfying the usual properties for an action. Here G is treated as a set, although the
properties involve the group operation on G. In the same way, a functor:
Φ̂ : G × C → C
satisfying analogous properties to φ̂, and where G is treated as a category as opposed to
a 2-category, is equivalent to an action of the 2-group G on the category C, as described
above (Theorem 3.4). From Φ̂ one obtains the description of the action as a transformation
groupoid internal to Cat. In this way we get a local symmetry picture for the global action
of a 2-group on a category, just like the global and local pictures that we had for the action
of an ordinary group.
We have been using two different understandings of a 2-group action on a category
C. The view where the 2-group is taken to be a group object in Cat is generally called
“internal”. By analogy, we may call the other “external”, i.e. in terms of a 2-functor
from the 2-category G to the 2-category Aut(C), whose sole object is C . The “internal”
description makes sense in any category where a group object can be defined, but describes
a 2-group action only in Cat. The “external” description always describes the action of
a 2-group, by its very definition. The existence of both internal and “external” views
of symmetry for a category relies on the fact that Cat is a closed category, that is, a
category with an “internal Hom”, so that the morphisms from A to B form an object
Hom(A, B) within the category. Thus Aut(C) can be seen as a category, equipped with
certain structure maps which make it into a 2-group.
Now, the internal, local symmetry picture associated with Φ̂ above gives rise to a
transformation groupoid in the same ambient category as our group object, and the
object it acts on. So for categories, we find that this local symmetry structure is most
naturally encoded by a category (indeed, it will be a groupoid) internal to Cat itself. Such
a structure is called Cat-category, or often a double category (though to avoid ambiguity,
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1433
we will reserve this name for a distinct but equivalent structure). In particular, this
structure is not generally just a 2-category. An external view of it shows that it has two
different kinds of morphism, denoted horizontal and vertical, and higher morphisms which
naturally have the shape of squares.2 We will show in Section 4 that there are several
complementary perspectives for viewing this structure, all highlighting particular features
of the 2-group action. The groupoid internal to Cat perspective has a dual description as a
category internal to Gpd (the category of groupoids), and a third, symmetric perspective
views the structure as a double category (becoming a double groupoid when the acted-on
category C is a groupoid). Thus our treatment here demonstrates that there is extra
subtlety, through these “variations in shape”, when we try to extend the concept of
symmetry to higher structures.
1.2. Motivation from higher gauge theory. Our motivation for the present study
comes from geometry, although this will play no direct role in this paper. We will,
however, describe a little of this motivation here, and in a subsequent paper we will give
more details about the example which prompted us to study this question.
The example we had in mind comes from higher gauge theory (HGT) [3]. This is a
general program of developing analogs to concepts in gauge theory, for higher-categorical
groups. It provides generalizations of, among other things, the theory of connections on
bundles over manifolds. These analogs include the study of connections on (non-abelian)
gerbes [14, 23, 20, 19, 27]. These entities have been the study of considerable interest in
recent years, particularly since they can be used to describe theories in which one can
naturally describe the parallel transport of extended objects such as (one-dimensional)
strings or (higher-dimensional) “membranes”.
Behind the current paper is a desire to better understand the symmetries of the “moduli spaces” for such theories. That is, in ordinary gauge theory, given a manifold M and
a Lie group G, there is a space of all connections on a given principal G-bundle over M .
Furthermore, there is a group which acts on this space, the group of all gauge transformations. There is a local picture of this symmetry, in which both connections and gauge
transformations are described in terms of certain forms and functions on M . The relation
between the global symmetries and this local picture is well-understood. The local picture
describes a groupoid, whose objects are the connections, coming from the global group
action.
In the case of connections on gerbes, there is also a well-understood local picture
in terms of forms and functions. However, these naturally form something more complex than a groupoid. The relationship between this local picture and a notion of global
symmetries is less well-established. In our subsequent paper, we will show a simple example of how the local picture arises from a global symmetry 2-group acting on a category which plays the role of the moduli space of connections on gerbes over M . For
even higher-categorical structures, known as n-gerbes, one can describe the transport of
2
The shape of higher morphisms is a typical issue in higher-categorical constructions: see e.g. [10, 18]
for discussion of variants of higher categories.
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JEFFREY C. MORTON AND ROGER PICKEN
higher-dimensional objects, or n-branes. This naturally has connections to string theory
[26, 2], and to other areas in theoretical physics.
Our main specific motivation within HGT relates to work of Yetter [29], and Martins
and Porter [21], extending a certain field theory (the Dijkgraaf-Witten model) to 2-groups.
One of the ultimate goals in our project is to reproduce for these theories a groupoid-based
approach to such a model [22], using the double groupoids we find here, relating these
field theories to the representations of the double groupoids.
While the motivation from HGT is of interest to us, it is not necessary to understand
the present paper. For the moment, we are simply interested in understanding better
how the relationship between local and global symmetries applies when the entity whose
symmetries are involved happens to be a category. Since the notion of symmetry is one
of the most fundamental, and categories have proven to be a vital and important part of
modern mathematics, the present study should be of interest well beyond the context of
our original example, and applicable in a variety of areas.
1.3. Results and outline. The main results of this work are structural theorems
relating to the action of a strict 2-group G on a category C, and the different ways in
which this action may be described.
We begin in Section 2 with a discussion of (strict) 2-groups, and various equivalent
definitions which are useful in talking about them. This includes the correspondence
between 2-groups and crossed modules and the definition of the double groupoid associated
to the 2-group G, which is both an introduction to the notion of double category and a
useful 2-dimensional diagrammatic tool for 2-group calculations.
This is followed by Section 3 which defines both the global and local notions of a
2-group G acting on a category C (see the first part of the introduction), i.e. Definitions
3.2 and 3.3, and shows the equivalence between the two definitions in Theorem 3.4. This
is followed in subsection 3.6 by a basic example, namely the adjoint action of a 2-group
on itself, analogous to the adjoint action of a group on itself by conjugation. (In the
forthcoming paper on higher gauge theory, we show that the adjoint action of G on itself
describes higher gauge theory for the simple case when the manifold is the circle, S 1 .)
In our main Section 4, we start by describing the construction of the transformation
groupoid associated to a group action on a set, in a diagrammatic form, which captures
the description of this groupoid in terms of elements. In subsection 4.2, repeating this
construction in Cat gives a groupoid internal to Cat, as shown in Theorem 4.4. We
develop a few different points of view on this internal groupoid in subsection 4.5, first
taking a transposed internal view, giving rise to a category internal to Gpd, the category
of groupoids (Theorem 4.8), and then giving a symmetrical definition in terms of a double
category (Theorem 4.10). The transposed viewpoint turns out to have close connections
to ordinary transformation groupoids for certain group actions. Further aspects of these
viewpoints are developed in subsection 4.12, and in particular we show that there are
two 2-categories associated to the transformation double category, namely the horizontal
2-category and the vertical 2-category, Theorems 4.16 and 4.17. Our standard example,
that of the adjoint action of a 2-group on itself, gives a concrete case throughout, which
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1435
we develop in detail in 4.18.
2. 2-Groups and 2-Group Actions on a Category
In this paper, we will be describing certain actions of 2-groups. The structures known as
2-groups are also sometimes called categorical groups, gr-categories or groupal groupoids.
Moreover, all of these can be shown to be equivalent to crossed modules (of groups).
As the abundance of terminology suggests, there are several conceptually different, but
logically equivalent, definitions which can be given for these structures. We will primarily
use the term “2-group”, but in fact it will be useful for our discussion to be able to
move back and forth between the different definitions. In Section 2.1 we will recall those
definitions which will be used, and note how they are equivalent.
This prepares the ground for section 3, where we will consider from two of these
points of view how 2-groups can act on categories, and the “transformation” structure
which results. This will be the analog of the transformation groupoid associated to a
group action on a set.
Note that in the following, to avoid cumbersome notation, we use the notation X(0)
and X(1) to denote, respectively, the objects and morphisms of a category X, and similar
notation for the object and morphism maps of a functor. For a 2-category X, when
appropriate, we denote the 2-morphisms by X(2) .
2.1. 2-Groups, Categorical Groups, and Crossed Modules. Here we lay out
three definitions of equivalent structures: 2-groups, categorical groups, and crossed modules. In this section, we will be careful to distinguish the three, but in the rest of the paper
we will generally use the term “2-group”. The main point of this section is to highlight
the well-known result that there is an equivalence between the three definitions.
2.1.1. 2-Groups as 2-Categories. One very useful definition of a 2-group is motivated
by analogy to the definition of a group as a (small) one-object category whose morphisms
are all invertible. In this case, the elements of the group are the set of morphisms of the
category, and the group multiplication is the composition.
This highlights the way the definition of a category generalized the notion of “composition” which begins with algebraic gadgets such as groups. One might equally well define
a (small) category in algebraic language as a unital semigroupoid. Taking the definition
of category as more basic gives a group as a special case. While of course not the standard
definition of a group, it is at least straightforward to generalize to “higher” groups in the
sense of higher category theory.
(We will assume some familiarity with 2-categories in what follows, though for readers
who are not so familiar we suggest the succinct note by Leinster [17] as a starting point,
and the more comprehensive survey by Lack [16] and Chapter 7 of Borceux [4] for more
detail.)
2.2. Definition. A 2-group G is a 2-category with one object (G (0) = {?}), for which
all 1-morphisms γ ∈ G (1) and 2-morphisms χ ∈ G (2) are invertible.
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JEFFREY C. MORTON AND ROGER PICKEN
Note that in the case of χ ∈ G (2) , we require invertibility with respect to both horizontal
and vertical composition.
Let us unpack this definition more explicitly.
Suppose G is a 2-group. There are various 1-morphisms from ? to itself, which have a
composition operation, since all such morphisms have matching source and target:
?o
γ1
?o
γ2
? o γ1 ◦γ2 ?
? =
(1)
Composition of morphisms is the multiplication, which is therefore associative, and every
morphism must have an inverse by the definition of a 2-group, so (G (1) , ◦) forms a group.
Moreover, there are 2-morphisms between the 1-morphisms. They have both a horizontal and a vertical composition. The horizontal composition we denote ◦, since this
is the same direction as the composition of 1-morphisms. The vertical composition we
denote ·. The two compositions must be compatible, in the sense that the following
composite is well-defined:
γ30
γ3
?\o
χ2
γ2
?\o
χ1
γ1
0
χ2
γ20
?
(2)
χ01
γ10
This is expressed as the “interchange law”
(χ1 · χ2 ) ◦ (χ01 · χ02 ) = (χ1 ◦ χ01 ) · (χ2 ◦ χ02 )
(3)
Again, these 2-morphisms are assumed to be invertible, so it follows that (G (2) , ◦)
forms a group. Note that we do not say that (G (2) , ·) forms a group, since · may not
be defined for all pairs of 2-morphisms, unless they have compatible source and target
1-morphisms.
However, one can find a group structure using vertical composition of 2-morphisms,
upon choosing a given 1-morphism γ, and considering Hom(γ, γ), the collection of 2morphisms from γ to itself. This is a group with operation ·, since 2-morphisms are
invertible under ·.
(Note that Hom(γ, γ) does not necessarily close under ◦, unless γ = 1G . In that case,
Hom(1G , 1G ) is an abelian group by the so-called Eckmann-Hilton argument, which uses
the interchange property (3) to show that horizontal and vertical composition agree, and
define an abelian group.)
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1437
We have noted the composition operations for both types of morphisms. It is also
significant that there is also a straightforward interaction between the two types, namely
whiskering. That is, a 1-morphism can act on a 2-morphism, either on the left or the
right, where it acts by composition with an identity 2-cell. That is, by convention we say:
γ1
?o
γ
?\
χ
γ◦γ2
? =
?\
Idγ ◦χ
γ2
γ◦γ1
?
(4)
and similarly for whiskering on the right.
Again, all these remarks are only unpacking what is implied by the definition of a
2-category with one object and invertible 1- and 2-morphisms. We will want to use this
definition occasionally, particularly when describing 2-group actions. However, for the
most part is will be useful to think of 2-groups in a way which is more amenable to
explicit calculations. The first step toward this is the definition of a categorical group,
and then even more explicit is the presentation in terms of a crossed module. We recall
these next.
2.2.1. Categorical Groups. Another natural approach to defining a “higher” analog
of a group is based on the view that a group is a “group object in Set”. A group object
in a category C with finite products (in particular, C = Set or C = Cat with Cartesian
products) is an object G ∈ C, equipped with structure maps such as the multiplication
map m : G × G → G satisfying the same axioms as those for a group. These axioms
can be expressed as commuting diagrams which make sense in any category with finite
products and specialize to the usual axioms in (Set, ×).
2.3. Definition. A (strict) categorical group is a (strict) group object in the monoidal
category (Cat, ×).
We remark here that there is a more general concept of (weak) categorical groups, but
we will restrict our attention to the strict case in the present paper. See Baez and Lauda
[1] for an exposition of the weak case.
That is, categorical groups are (small) categories G, equipped with functors ⊗ : G ×
G → G, and inv : G → G satisfying the group axioms.
Since Cat is, in fact, a monoidal 2-category (in which the monoidal product on objects
is the cartesian product of categories), our definition specifies that the functors ⊗ and inv
satisfy the axioms for a group strictly. That is, the usual equations such as associativity
of multiplication are still equations, rather than 2-isomorphisms. The corresponding weak
notion, in which equations are replaced by such 2-isomorphisms, necessarily has additional
coherence conditions they must satisfy.
Notice that the “group multiplication” functor, which we have written as ⊗, satisfies the properties for a monoidal product, as the notation suggests. Indeed, a (strict)
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JEFFREY C. MORTON AND ROGER PICKEN
monoidal category is simply a monoid object in Cat. A (strict) categorical group is therefore a strict monoidal category with inverses. That is, every object and morphism has an
inverse with respect to ⊗.
It is standard, and easy to see, that any 2-group as defined in Section 2.1.1 gives rise
to a categorical group, and vice versa. We describe the correspondence here to settle
notation.
2.4. Definition. Given a 2-group G, the categorical group (C(G), ⊗) associated to G is
defined as follows:
• Objects: C(G)(0) = G (1) (objects are morphisms of G)
• Morphisms: C(G)(1) = G (2) (morphisms are 2-morphisms of G)
• The composition of morphisms of C(G) is vertical composition of 2-morphisms in G
• The monoidal product ⊗ of C(G) is:
– On C(G)(0) , same as composition in G (1)
– On C(G)(1) , same as horizontal composition in G (2)
• The inverse functor inv : C(G) → C(G) is given, for each object or morphism, by
the inverse or vertical inverse of the corresponding 1- or 2- morphism in G
Given a categorical group (G, ⊗), the 2-group ?//G has:
• Object: just one, ?
• Morphisms: (?//G)(1) = G(0) , the objects of G
• 2-Morphisms: (?//G)(2) = G(1) , the morphisms of G
• Composition for (?//G)(1) and horizontal composition for (?//G)(2) are ⊗(0) and ⊗(1)
from G respectively
• Vertical composition for (?//G)(2) is composition for G(1)
It is well known, and the unfamiliar reader may easily check, that these two correspondences give an equivalence of the two definitions. For example, the properties of
invertibility for ?//G follow from the existence of inv in (G, ⊗), and the fact that it satisfies group axioms. It is similarly easy to check that monoidal functors between categorical
groups correspond to 2-functors between 2-groups. For example, the interchange law for
2-groups is expressed, in categorical groups, as the fact that the monoidal product is
functorial. The convention for whiskering in a 2-group is easily translated to the usual
convention for the monoidal product of an object with a morphism.
Another equivalent presentation is a result of the fact that there is a correspondence
between groups internal to categories, and categories internal to groups (that is, categories
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1439
whose sets of objects and sets of morphisms each have the structure of a group, and
where source, target, and composition operations are all group homomorphisms). This
equivalence follows naturally, since given a categorical group, the structure maps, such
as ⊗, are functors which satisfy the group axioms. The object and morphism maps for
these functors therefore each separately satisfy the same axioms. Thus, the objects of a
categorical group form a group, as do the morphisms of the categorical group. Thinking
of these separate group structures naturally leads to the third of the definitions we will
use in this paper, namely crossed modules.
2.4.1. Crossed Modules. A well-known theorem due to Brown and Spencer [8] says
that (strict) 2-groups are classified by crossed modules. See also [5] for further background.
2.5. Definition. A crossed module consists of (G, H, B, ∂), where G and H are groups,
G B H is an action of G on H by automorphisms and ∂ : H → G is a homomorphism,
satisfying the equations:
∂(g B η) = g∂(η)g −1
(5)
and
∂(η) B ζ = ηζη −1
(6)
We describe the correspondence with categorical groups, which will be the form we
make the most use of, though of course the correspondence with 2-groups follows immediately as well:
2.6. Definition. The categorical group (G, ⊗) given by (G, H, B, ∂) has:
• Objects: G(0) = G
• Morphisms: G(1) = G × H, with source and target maps
s(g, η) = g
(7)
t(g, η) = ∂(η)g
(8)
Idg = (g, 1H )
(9)
(∂(η)g, ζ) ◦ (g, η) = (g, ζη).
(10)
and
• Identities:
• Composition:
• Monoidal product: given on objects by g1 ⊗ g2 = g1 g2 and on morphisms by
(g1 , η) ⊗ (g2 , ζ) = (g1 g2 , η(g1 B ζ))
which corresponds to the horizontal composition of 2-morphisms in G.
(11)
1440
JEFFREY C. MORTON AND ROGER PICKEN
The theorem alluded to above asserts that any strict 2-group is equivalent to one of
this form. We will usually assume that any 2-group we use from now on is presented by
a crossed module, and use the corresponding notation for explicit calculations.
The structure of a category which lies behind the definition of a crossed module is
somewhat disguised. Much of it is included in the fact that G and H are groups. The
explicit action of G on H captures the notion of whiskering, remarked upon above, and
the projection onto the first factor, together with the “boundary” map ∂, determines the
source and target structure maps for the categorical group. However, for example, the
vertical composition of 2-morphisms is not explicitly mentioned. Full details of how it
and other categorical group structures can be reconstructed from a crossed module are
well recorded elsewhere.
2.7. Double Groupoid of a 2-Group. A useful construction for calculations later on
is the double groupoid of G.
Double categories were introduced by Ehresmann [12, 11], and can be defined in several
equivalent ways, three of which are noted by Brown and Spencer [7]. The most pertinent
later in this paper is that they are categories internal to Cat, the category of all categories.
This terse definition is the most common in current use, but is somewhat opaque, and
obscures the basically symmetric nature of these structures.
One can also describe a (small) double category in a more manifestly symmetric way.
Although the two definitions are equivalent, they are superficially rather different. In
Section 4.5 we will use both points of view to describe the structure which it is the goal of
this paper to construct. Therefore, to avoid confusion, we will reserve “double category”
for the symmetric view, and use “Cat-category” to mean the equivalent structure seen
as a category in Cat.
We will not give a full definition of “double category” here, though, since it is equivalent
to and can be deduced from the terse definition, but it is intuitively useful to see a double
category D as consisting of:
• a set of objects O
• two sets H and V of morphisms (we use calligraphic letters here to avoid confusion
with the group H), denoted horizontal and vertical, together with structure maps
making them into horizontal and vertical categories with objects O
• a set of squares S which form the morphisms of two category structures whose
objects are H and V respectively
The category structures on the squares S satisfy some extra properties making them compatible with the category structures on H and V. Intuitively, they say that “horizontal”
structure maps commute with “vertical” structure maps. For example, one can take a horizontal composite of two squares σ1 and σ2 , and then take the morphism sv (σ1 ◦h σ2 ) ∈ H
which is its vertical source. This is the same as sv (σ1 ) ◦h sv (σ2 ), the horizontal composite
of the vertical sources of each square. There are many other conditions of this form, such
as the “interchange law” for composition of squares, which has the same form as (3).
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1441
A double groupoid is a double category in which all morphisms and squares are invertible. Further discussion of double groupoids is found in Brown and Spencer [7], while
an interesting summary of general folklore about this symmetric definition and variations
on it can be found online [25].
The above intuition essentially justifies a graphical notation which makes 2-group
calculations more straightforward. This notation has been extensively used elsewhere, see
e.g. [5], and we now describe it for reference.
2.8. Definition. Given a 2-group G, the double groupoid of G, denoted D(G), is the
double groupoid with:
• A unique object of D(G): O = G (0) = {?}
• Horizontal and vertical morphisms of D(G) are H = V = G (∞)
• Squares of D(G) are given by all squares of the form
g3
g4
η
g2
g1
where gi ∈ G, i = 1, . . . , 4, η ∈ H and ∂(η) = g1 g2 g3−1 g4−1
• horizontal and vertical composition of squares is given by
g7
g3
g3 g7
g4
η1 g2 η2
g1
η1 (g4 g3 g2−1 ) B η2
g6 = g4
g5
g3 g7
g6 = g4 (g1 B η2 )η1 g6
g1 g5
g1 g5
where the two expressions for the H element in the horizontal compositions are the
same, using ∂(η1−1 ) B η2 = η1−1 η2 η1 , and
g3
g4
η1
g2
g1
g5
η2
g3
= g5 g4 η2 (g5 B η1 ) g7 g2
g7
g6
g6
and these operations satisfy the interchange law, i.e. the equality of evaluating a
2 by 2 array of squares in two different ways (first horizontal and then vertical
composition, or vice-versa).
In Ehresmann’s terminology, this is the double category of quintets of the bicategory
G: the term refers to the fact that squares are determined by five pieces of data: the four
1-morphisms which are its edges, and the 2-morphism which fills the square.
1442
JEFFREY C. MORTON AND ROGER PICKEN
2.9. Remark. Due to the interchange law there is a consistent evaluation of any rectangular array of squares, which is independent of the order in which the horizontal and
vertical multiplications are performed. Also squares in D(G) have horizontal and vertical
inverses, which are respectively given by (for the square in the definition):
g3−1
g2
η −h
g1−1
g3−1
g4 = g2
g1−1 B η −1
g1
g4−1
g4
η −v
g1
g2−1 = g4−1
g3
g1−1
g4−1 B η −1
g2−1
g3
In particular, the double groupoid D(G) contains a copy of G by considering only
the horizontal morphisms, and the squares for which the vertical source and target are
identities. Thus, a typical square of this sort is of the following form:
g1
η
g2
where here, and henceforth, any unlabelled edge or square is taken to be labelled with
the identity of the corresponding group. This can be identified with a 2-morphism in the
2-group G, namely:
g1
?\
η
?
(12)
g2
or equivalently, with the morphism (g1 , η) in the categorical group.
3. Actions of 2-Groups on Categories
Our aim is to describe actions of a 2-group G on a category C. There are two equivalent
ways of describing these, depending on whether one views G as a 2-group or a categorical
group. In this section, we will outline these two views and see the relation between them.
Then we will consider a natural example, namely the adjoint action of a categorical group
on its own underlying category.
There are two different, but equivalent, definitions of group actions on a set, and both
will be relevant for us. To begin with, consider a group G, seen as a one-object category
whose morphisms are all invertible. Then a G action φ on a set X may be described as a
functor
φ : G → Set
(13)
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1443
where X = φ(?) is the image of the unique object of G, i.e. the image of φ is End(X), the
full subcategory of Set with the single object X (in fact, since G is a group, the image is
Aut(X), consisting of only the invertible endomorphisms of X).
Next we show how to construct the transformation groupoid in a way that naturally
generalizes to the 2-group case, by regarding the group G as a set equipped with a
multiplication map m : G × G → G and an inverse inv : G → G, satisfying the group
axioms. Thus we have the familiar definition of an action on a set X as a function
φ̂ : G × X → X
(14)
This function is related to φ by φ̂(γ, x) = φγ (x). Functoriality of φ means, in particular,
that φ̂ satisfies a compatibility condition with the multiplication map m : G × G → G,
namely that the following commutes:
G×G×X
IdG ×φ̂
m×IdX
/
G×X
G×X
/
φ̂
(15)
φ̂
X
φ̂ also satisfies a unit condition:
φ̂(1, x) = x, ∀x ∈ X
(16)
This definition is, of course, equivalent to the point of view of an action as a functor.
First it determines φ(?) = X. The two definitions are then related by turning a function
G → Hom(X, X) into an X-valued function of G × X, taking (γ, x) to φγ (x). (The
term for this in logic is “uncurrying”, while “currying” denotes the process which turns
a function of n variables into a chain of n one-variable functions, each one returning the
next function in the chain).
3.1. Actions of 2-Groups on Categories. Next we extend the two viewpoints of
the previous introduction for the action of a group on a set to the action of a 2-group on
a category. First, by analogy with (13), regarding a 2-group as a 2-category, an action
will be a 2-functor into the 2-category Cat.
3.2. Definition. A 2-group G acts (strictly) on a category C if there is a (strict) 2functor:
Φ : G → Cat
(17)
whose image lies in End(C), the full sub-2-category of Cat with the single object C.
Thus Φ(∗) = C, and on 1- and 2-morphisms Φ is given by assignments:
• for each γ ∈ G (1) we have the endofunctor Φγ : C → C, acting as
f
Φγ (f )
(x → y) 7→ (Φγ (x) −→ Φγ (y))
1444
JEFFREY C. MORTON AND ROGER PICKEN
• for each 2-morphism (γ1 , χ) ∈ G (2) , we have the natural transformation Φ(γ1 ,χ) :
Φγ1 → Φγ2 , where γ2 = ∂(χ)γ1 , given by assignments (C(0) 3 x) 7→ (Φ(γ1 ,χ) (x) ∈
C(1) ) satisfying the naturality condition
Φγ1 (x)
Φγ1 (f )
/ Φγ
1
(y)
Φ(γ1 ,χ) (y)
Φ(γ1 ,χ) (x)
Φγ2 (x)
Φγ2 (f )
(18)
/ Φγ
2
(y)
These assignments must satisfy the conditions to be a strict 2-functor, i.e. they preserve
all vertical and horizontal compositions and identities, which here means:
Ver
1) Φ(γ2 ,χ2 ) (x) ◦v Φ(γ1 ,χ1 ) (x) = Φ(γ1 ,χ2 χ1 ) (x) where γ2 = ∂(χ1 )γ1
2) Φ(γ,1) (x) = idΦγ (x)
Hor
1) Φγ1 ◦ Φγ3 = Φγ1 γ3 , Φ1 = idC
2) Φ(γ1 ,χ1 ) ◦h Φ(γ3 ,χ2 ) = Φ(γ1 γ3 ,χ1 (γ1 Bχ2 )) .
Writing out explicitly the horizontal composition of natural transformations
on the l.h.s., this final condition becomes:
Φ(γ1 ,χ1 ) (Φγ4 (x)) ◦ Φγ1 (Φ(γ3 ,χ2 ) (x)) = Φ(γ1 γ3 ,χ1 (γ1 Bχ2 )) (x)
where γ4 = ∂(χ2 )γ3 . Here the underlying array of squares in D(G) is
γ1
γ3
χ1
χ2
γ2
γ4
(19)
The second viewpoint for a 2-group to act on a category is to regard the 2-group G as
a categorical group, i.e. a monoidal category, and proceed by analogy with (14), (15).
3.3. Definition. A strict action of a categorical group G on a category C is a functor
Φ̂ : G × C → C satisfying the action square diagram in Cat (strictly):
G×G×C
IdG ×Φ̂
⊗×IdC
G×C
/
G×C
/
Φ̂
(20)
Φ̂
C
and the unit condition
Φ̂(1, x) = x, ∀x ∈ C(0)
(21)
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1445
Thus, from functoriality, we have the conditions:
Φ̂((γ2 , χ2 ), g) ◦ Φ̂((γ1 , χ1 ), f ) = Φ̂((γ1 , χ2 χ1 ), g ◦ f )
Φ̂((γ, 1H ), idx ) = idΦ̂(γ,x)
(22)
(23)
where, in (22), γ2 = ∂(χ1 )γ1 . The action square diagram corresponds to the equations
(on objects and morphisms respectively):
Φ̂(γ1 γ3 , x) = Φ̂(γ1 , Φ̂(γ3 , x))
(24)
Φ̂((γ1 γ3 , χ1 (γ1 B χ2 )), f ) = Φ̂((γ1 , χ1 ), Φ̂((γ3 , χ2 ), f )).
(25)
Here the underlying array of squares in D(G) is (19).
The following theorem shows that these two viewpoints are equivalent.
3.4. Theorem. A strict 2-functor Φ : G → End(C) is equivalent to a strict action
functor Φ̂ : G × C → C.
Proof. Given Φ, set: Φ̂(γ, x) := Φγ (x) and
Φ(γ1 ,χ) (y) ◦ Φγ1 (f )
Φ̂((γ1 , χ), f )/
/ Φγ (y) )
Φ̂(γ2 , y) ) := ( Φγ1 (x)
( Φ̂(γ1 , x)
2
(26)
By the naturality condition (18), this is also equal to
Φγ2 (f ) ◦ Φ(γ1 ,χ) (x)
/ Φγ (y) )
( Φγ1 (x)
2
(27)
Now functoriality for Φ̂ follows from putting together four naturality squares (18) in the
obvious way, and then using functoriality of Φγ horizontally and the first Ver condition
vertically.
The first action square equation (24) and the unit condition (3.3) are immediate. The
second action square equation (25) follows from:
Φ̂((γ1 γ3 , χ1 (γ1 B χ2 )), f ) = Φ(γ1 γ3 ,χ1 (γ1 Bχ2 )) (y) ◦ Φγ1 γ3 (f )
= Φ(γ1 ,χ1 ) (Φγ4 (y)) ◦ (Φγ1 (Φ(γ3 ,χ2 ) (y)) ◦ Φγ1 (Φγ3 (f )))
= Φ(γ1 ,χ1 ) (Φγ4 (y)) ◦ (Φγ1 (Φ(γ3 ,χ2 ) (y)) ◦ Φγ3 (f )))
= Φ̂((γ1 , χ1 ), Φ(γ3 ,χ2 ) (y) ◦ Φγ3 (f ))
= Φ̂((γ1 , χ1 ), Φ̂((γ3 , χ2 ), f ))
using the first Hor condition for Φ and associativity in the second equality.
Conversely, given Φ̂, set Φγ (x) := Φ̂(γ, x) (on objects),
Φγ (f )
Φ̂((γ, 1H ), f )
/ Φ(γ, y) )
/ Φγ (y) ) := ( Φ̂(γ, x)
( Φγ (x)
1446
JEFFREY C. MORTON AND ROGER PICKEN
(on morphisms), and on 2-morphisms the natural transformation Φ(γ1 ,χ) : Φγ1 → Φγ2 ,
where γ2 = ∂(χ)γ1 , is given by:
( Φγ1 (x)
Φ(γ1 ,χ) (x)
/
Φ̂((γ1 , χ), idx )
/ Φ̂(γ2 , x) )
Φγ2 (x) ) := ( Φ̂(γ1 , x)
(28)
Now, the Hor properties for Φ follow in a straightforward manner using the functoriality
conditions (22), (23) for Φ̂. The first Ver property for Φ is immediate, and the second
Ver property follows from:
Φ(γ1 ,χ1 ) (Φγ4 (x)) ◦ Φγ1 (Φ(γ3 ,χ2 ) (x)) = Φ̂((γ1 , χ1 ), idΦ̂(γ4 ,x) ) ◦ Φ̂((γ1 , 1H ), Φ̂((γ3 , χ2 ), idx ))
= Φ̂((γ1 , χ1 ), Φ̂((γ3 , χ2 ), idx ))
= Φ̂((γ1 γ3 , χ1 (γ1 B χ2 )), idx ))
= Φ(γ1 γ3 ,χ1 (γ1 Bχ2 )) (x)
where we use functoriality of Φ̂ (22) in the second equality and the second action square
condition (25) applied to f = idx in the penultimate equality.
It is convenient to introduce a succinct notation for a 2-group action analogous to the
usual notation g B x = φg (x) for a group action. There are actually three possible maps.
Since Φγ is a functor with both object and morphism maps, the objects of G act on
both the set of objects and the set of morphisms of C. Moreover, the action functor Φ̂
also has both object and morphism maps. The object map Φ̂(γ, −) is the same as the
object map for Φγ , by the above argument. However, the morphism map determines an
action of the morphisms of G on the morphisms of C, which is a different action again.
We will return to the relation between these three actions again in Corollary 4.14. For
the moment, is convenient to use the symbol I to denote all three, as follows.
3.5. Definition. If G is a 2-group classified by the crossed module (G, H, B, ∂), let the
notation I denote the following.
• Given γ ∈ G (0) = G and x ∈ C(0) , let
γ I x = Φγ (x) = Φ̂(γ, x)
(29)
• Given γ ∈ G (0) = G and f ∈ C(1) , let
γ I f = Φγ (f ) = Φ̂((γ, 1H ), f )
(30)
• Given (γ, χ) ∈ G (1) = G n H and (f : x → y) ∈ C(1) , let
(γ, χ) I f = Φ̂((γ, χ), f )
= Φ(γ,χ) (y) ◦ (γ I f )
= (∂(χ)γ I f ) ◦ Φ(γ,χ) (x)
(31)
The last two expressions are the same as those given in the proof of Theorem 3.4,
written in our new notation. It will be revisited in (66).
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1447
3.6. Example: Adjoint Action of 2-Groups. Here we want to describe the adjoint
action of a 2-group G on itself. This is the analog of the usual adjoint action of a group
G on itself by conjugation. The adjoint action is a functor
Φ : G → End(G)
(32)
Φγ (g) = γgγ −1
(33)
given by the property that
We want a 2-functor given “by conjugation”, insofar as this makes sense. In fact, as
we shall see, this is easy to do in the language of crossed modules, since the axioms (5)
and (6) for a crossed module (G, H, B, ∂) imply that the action B of G on H resembles
conjugation as much as possible. This will be made even clearer by using the square
calculus in the double groupoid of G, D(G).
In accordance with Definition 3.2, we take the action of G on itself to be given by a
2-functor from G to Cat with image in End(G), i.e. the “acting” G is regarded as a 2category, whilst the “acted on” G is regarded as a (monoidal) category. (This is analogous
to the situation for the adjoint action of a group G, regarded as a category, acting on G,
regarded as a set).
3.7. Definition. Suppose G is the 2-group given by a crossed module (G, H, B, ∂). Then
we define a strict 2-functor:
Φ : G → Cat
(34)
with image in End(G) in the following way. At the object level, Φ(∗) = G, where G on
the right is taken to be a category. For each morphism γ ∈ G (1)
Φγ : G → G
(35)
is a morphism of End(G), namely an endofunctor of G. Its object map is given by:
Φγ (g) = γgγ −1
(36)
Φγ (g, η) = (γgγ −1 , γ B η)
(37)
and its morphism map is given by
For each 2-morphism (γ, χ) ∈ G (2) there is a natural transformation from Φγ to Φ∂(χ)γ ,
given by:
Φ(γ,χ) (g) = (γgγ −1 , χ(γgγ −1 ) B χ−1 ))
(38)
In the action notation of Definition 3.5, the first two simply say that γ I g = γgγ −1
and γ I (g, η) = (γ I g, γ Bη). The third part of that definition will define (γ, χ) I (g, η),
which is not directly given by Φ. However, we can understand it using the calculus of
squares introduced in Section 2.7 for the double groupoid of quintets D(G) associated to
G.
1448
JEFFREY C. MORTON AND ROGER PICKEN
Represent morphisms (g1 , η) in the category G as squares, so that the action of Φγ on
such a square is given by:
g1
η
γ
Φγ
γ
g2
γg1 γ −1
γ −1
η
7→
g2
g1
=
γ −1
γBη
γ I g1
=
γBη
(39)
γ I g2
γg2 γ −1
where g2 = ∂(η)g1 . Likewise we can represent the morphism Φ(γ1 ,χ) (g) (which has no
direct analog in the action notation) as
γ1
g
χ
Φ(γ1 ,χ) (g) =
γ2
γ1−1
γ1 I g
= χ(γ1 I g) B χ−1
χ−h
g
(40)
γ2 I g
γ2−1
where γ2 = ∂(χ)γ1 .
It follows from (39) that Φγ (g1 , η) has the correct source and target. Φγ is a functor,
since it preserves identities, Φγ (g, 1H ) = (γgγ −1 , 1H ) (immediate), and composition, i.e.
Φγ (g2 , η2 ) ◦ Φγ (g1 , η1 ) = Φγ (g1 , η2 η1 ), where g2 = ∂(η1 )g1 , because of:
γ
g1
γ −1
γ
η1
γ
g2
γ −1
γ
g3
γ −1
η2 η1
=
η2
g1
γ
g3
γ −1
γ −1
It then follows immediately from (40) that Φ(γ1 ,χ) (g) has the correct source and target.
The naturality condition (18), i.e. Φγ2 (g1 , η) ◦ Φ(γ1 ,χ) (g1 ) = Φ(γ1 ,χ) (g2 ) ◦ Φγ1 (g1 , η), is the
equality:
γ1−1
γ1−1
γ1
g1
γ1
g1
χ
γ2
g1
χ−h
γ2−1
η
γ2
g2
η
=
γ1
g2
χ
γ2−1
γ2
γ1−1
χ−h
g2
γ2−1
which follows from the interchange law by evaluating the vertical compositions on both
sides first.
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1449
The first Ver condition, i.e. Φ(γ2 ,χ2 ) (g) ◦ Φ(γ1 ,χ1 ) (g) = Φ(γ1 ,χ2 χ1 ) (g) where γ2 = ∂(χ1 )γ1 ,
follows from the equality:
γ1
g
γ1−1
g
χ−h
1
γ2−1
χ1
γ2
χ2 χ1
=
χ−h
2
χ2
γ3
g
γ1−1
g
γ1
γ3
(χ2 χ1 )−h
g
γ3−1
γ3−1
The second Ver condition, i.e. Φ(γ,1H ) (g) = IdΦγ (g) , and the first Hor condition, i.e.
Φγ1 (Φγ3 (g)) = Φγ1 γ3 (g) and Φ1 (g) = g, are both immediate. Finally the second Hor
condition, i.e. Φ(γ, χ1 ) (Φγ4 (g)) ◦ Φγ1 (Φ(γ3 ,χ2 ) (g)) = Φ(γ1 γ3 ,χ1 (γ1 Bχ2 )) (g), corresponds to the
equality:
γ1
γ1
χ1
γ2
γ3
χ2
γ4
γ4
g
γ3−1
g
χ−h
2
γ4−1
g
γ4−1
γ1−1
γ1 γ3
γ1−1
χ−h
1
=
(γ1 γ3 )−1
g
(χ1 (γ1 B χ2 ))−h
χ1 (γ1 B χ2 )
γ2 γ4
g
(γ2 γ4 )−1
γ2−1
This follows from evaluating the two 2 by 2 arrays of squares without g on the left hand
side. Thus we have proved:
3.8. Lemma. The 2-group adjoint action of Definition 3.7 is a well-defined action in the
sense of Definition 3.2.
3.9. Remark. We can now display the functor Φ̂ of Definition 3.3, in terms of squares,
and therefore finish describing this action in the notation of Definition 3.5. Namely
(γ1 , χ) I (g1 , η) = Φ̂((γ1 , χ), (g1 , η)) is given by:
γ1
g1
γ −1
χ
η
χ−h
γ2
g2
γ2−1
γ1
=
χ
∂(χ)
g1
γ −1
η
γ1
g2
χ−1
γ1−1 ∂(χ)−1
The 5-square array on the right makes it clear that the 2-group adjoint action of (γ1 , χ) can
be regarded as the ordinary adjoint action of γ1 on edges, extended to squares labelled by
γ1 acting on squares, followed by the adjoint action of χ, in the sense of conjugation with
the square labelled χ. Evaluating the array and dropping the indices gives an algebraic
formula for this action:
(γ, χ) I (g, η) = (γgγ −1 , χ(γ B η)(γgγ −1 ) B χ−1 )
(41)
1450
JEFFREY C. MORTON AND ROGER PICKEN
4. Transformation Double Category for a 2-Group Action
In this section, we recall the construction of the transformation groupoid for a group action
on a set, and consider the analogous construction for a 2-group action on a category.
If we are given an action of a group on a set, there is a groupoid which corresponds
to it.
4.1. Definition. Given a group action φ : G → End(X), the transformation groupoid
X//G is the groupoid with:
• Objects: x ∈ (X//G)(0) = X
• Morphisms: (γ, x) ∈ (X//G)(1) = G × X, with source and target maps s(γ, x) = x,
and t(γ, x) = φγ (x)
• Composition: (γ 0 , φγ (x)) ◦ (γ, x) = (γ 0 γ, x)
It is clear that this is a groupoid, whose morphisms are invertible since G is a group.
The composition then encodes both the group multiplication (in the first component) and
the action (in the second component). Formally, the set P = (X//G)(1) ×(X//G)(0) (X//G)(1)
of composable pairs of morphisms in X//G is then given by the pullback square:
/
P
G×X
πX
G×X
/
(42)
φ̂
X
There is an obvious commuting diagram with G×G×X replacing P in the above diagram,
and hence, by the universal property of this pullback, there is a unique map from G×G×X
to P , given by:
(γ 0 , γ, x) 7→ ((γ 0 , φγ (x)), (γ, x))
(43)
which is clearly an isomorphism. In this way, the composition map from P to G × X
agrees with the map m × IdX in (15).
In the next subsections, we will develop a similar construction in Cat, which will give
a transformation double category for a categorical group action. As discussed in Section
2.7, we reserve the term “double category” for a certain symmetric point of view of this
structure. To construct it, however, we use an equivalent definition as a Cat-category.
4.2. Construction of the Transformation Cat-Category for a 2-Group
Action. The most obvious way to define an analog of the transformation groupoid in the
situation of a 2-group action comes by simply following the same constructions from an
ordinary group action, replacing G with G and X with C. Thus, one has action diagrams
in Cat.
Next we will construct a transformation groupoid as we did for group actions on sets,
but it will be internal to Cat, so we call it a transformation Cat-groupoid.
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1451
Thus, now one has a category (C//G)(0) of objects, and a category (C//G)(1) of morphisms. These, of course, have objects and morphisms of their own, but this fact is
invisible to the construction and becomes important only when we want to describe the
resulting structure concretely.
4.3. Definition. Given a 2-group G, a category C, and an action of G on C as in
Definitions 3.2 and 3.3, the transformation Cat-groupoid C//G is the groupoid internal
to Cat with:
• Category of objects: (C//G)(0) = C.
• Category of morphisms: (C//G)(1) = G × C, with
– Source functor s = πC : G × C → C
– Target functor t = Φ̂ : G × C → C
– Identity inclusion functor e = 1G × IdC : C → G × C
– Inverse functor inv = (invG , t) : G × C → G × C
• Category of composable pairs: P, given by the pullback diagram
/
P
G×C
πC
G×C
/
(44)
Φ̂
C
• Composition functor: given by the action square (20)
The situation is entirely analogous to the transformation groupoid for an ordinary
group G, as is seen easily by considering the effect on objects. (Thus the inverse functor,
on objects, takes (γ, x) 7→ (γ −1 , γ I x), just as with an ordinary transformation groupoid.)
Analogously with the situation for groups, we can think of (C//G)(1) as a semidirect
product 2-group; and there is a canonical isomorphism between G × G × C and P. We
should check that this definition makes sense.
4.4. Theorem. The transformation Cat-groupoid C//G is a well-defined groupoid internal to Cat.
Proof. To confirm this, we must check that the source, target, composition, identity
inclusion and inverse functors are well defined, and satisfy the usual properties for a
groupoid, primarily associativity and the left and right unit laws.
The fact that the source and target maps are functors is obvious, since they are just
the projection from a product, and Φ̂ respectively. The first is necessarily a functor, while
the second is a functor by definition.
1452
JEFFREY C. MORTON AND ROGER PICKEN
Recall our notation:
γ I x = Φγ (x) = Φ̂(γ, x)
(γ, χ) I f = Φ̂((γ, χ), f )
(45)
and make explicit the source, target and composition functors at the morphism level as
follows:
s((γ, χ), f ) = f
t((γ, χ), f ) = (γ, χ) I f
(46)
((γ1 , χ1 ), (γ3 , χ2 ) I f ) ◦ ((γ3 , χ2 ), f ) = ((γ1 γ3 , χ1 (γ1 B χ2 )), f )
(47)
(see (25)). The target of the composition is well-defined due to the commutativity of (20):
(γ1 , χ1 ) I ((γ3 , χ2 ) I f ) = (γ1 γ3 , χ1 (γ1 B χ2 )) I f
(48)
To show the functoriality of composition, we refer to the following array of squares in
D(G), underlying the calculation:
γ1
γ3
χ1
χ2
γ2
χ3
γ4
χ4
γ5
γ6
(49)
We denote composition in G × C by ◦¯ and composition in C by ◦C , to distinguish them
from the composition functor ◦. Then we have:
((γ4 , χ4 ), g) ◦¯ ((γ3 , χ2 ), f ) = ((γ3 , χ4 χ2 ), g ◦C f )
((γ2 , χ3 ), (γ4 , χ4 ) I g) ◦¯ ((γ1 , χ1 ), (γ3 , χ2 ) I f ) = ((γ1 , χ3 χ1 ), ((γ4 , χ4 ) I g) ◦C
((γ3 , χ2 ) I f ))
= ((γ1 , χ3 χ1 ), (γ3 , χ4 χ2 ) I (g ◦C f ))
where we use (22) in the final equation. We also have:
((γ2 , χ3 ), (γ4 , χ4 ) I g) ◦ ((γ4 , χ4 ), g) = ((γ2 γ4 , χ3 (γ2 B χ4 )), g)
((γ1 , χ1 ), (γ3 , χ2 ) I f ) ◦ ((γ3 , χ2 ), f ) = ((γ1 γ3 , χ1 (γ1 B χ2 )), f )
Thus it remains to show the equality of two expressions:
((γ2 γ4 , χ3 (γ2 B χ4 )), g) ◦¯ ((γ1 γ3 , χ1 (γ1 B χ2 )), f ) = ((γ1 γ3 , χ3 (γ2 B χ4 )χ1 (γ1 B χ2 )), g ◦C f )
and
((γ1 , χ3 χ1 ), (γ3 , χ4 χ2 ) I (g ◦C f )) ◦ ((γ3 , χ4 χ2 ), g ◦C f ) = ((γ1 γ3 , χ3 χ1 γ1 B (χ4 χ2 )), g ◦C f ).
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1453
This equality is immediate from the interchange law in D(G) applied to the array (49).
Associativity of the composition functor follows from the fact that the horizontal
composition of three squares in D(G) has a unique evaluation:
γ1
γ3
γ5
χ1
χ2
χ3
γ2
γ4
γ6
(50)
The identity inclusion e is clearly a functor, and satisfies:
((γ, χ), f ) ◦ ((1G , 1H ), f ) = ((γ, χ), f )
((1G , 1H ), (γ, χ) I f ) ◦ ((γ, χ), f ) = ((γ, χ), f )
where in the first equation we have a composable pair, since (if the target of f is y):
t((1G , 1H ), f ) = (1G , 1H ) I f = Φ(1G ,1H ) (y) ◦ Φ1G (f ) = Φ(1G ,1H ) (y) ◦ f = f
using (26) in the second equality, functoriality of Φ in the third equality and property F2
(see Def. 3.2 ) in the final equality.
Finally, the inverse functor assigns, to every ((γ, χ), f ) in the category of morphisms,
its inverse ((γ −1 , χ−h ), (γ, χ) I f ), where we note that (γ −1 , χ−h ) = (γ −1 , (∂(χ)γ)−1 B
χ−1 ) = (γ −1 , γ −1 B χ−1 ). This follows immediately from (47). Functoriality of the inverse
functor follows easily from (22) and functoriality of the horizontal inverse in D(G).
This construction of a Cat-category has a slight drawback, which is the apparent
asymmetry of its definition. In the next section we will address this issue.
4.5. The Transformation Double Category. The construction we have given in
the previous section naturally produces a groupoid internal to Cat. However, the view
of a double category as an internal category in Cat obscures the underlying symmetry of
this structure, and the structure in the “transverse” direction to the categories of objects
and morphisms.
As we remarked in Section 2.7, Cat-categories are equivalent to a structure defined
in a more symmetric way, having two types of morphisms (horizontal and vertical), and
squares. The properties can be deduced from the equivalence: for example, the “interchange law”, which says that mixed horizontal and vertical composites can be taken in
any order, amounts to the functoriality of the composition ◦.
Now, the definition of double categories is symmetric under exchanging the roles of
“horizontal” and “vertical” morphisms in our diagrams of squares. One can reflect all
squares in a diagonal, and obtain another double category, which we call the transpose of
the first double category. The resulting structure, naturally, still has an interpretation as
a category internal to Cat.
More technically, we have the following:
1454
JEFFREY C. MORTON AND ROGER PICKEN
4.6. Definition. If D is an internal category in Cat, then let the transpose of D,
e be the internal category in Cat defined by the following:
which we denote D,
• The category of objects has
– Objects: the objects of the category D(0)
– Morphisms: the objects of the category D(1)
• The category of morphisms has
– Objects: the morphisms of the category D(0)
– Morphisms: the morphisms of the category D(1)
• The identity inclusion, source and target, and composition functors (e
e, se, e
t, e
◦) have
(0)
as object maps the corresponding structure maps from the category D , and as
morphism maps the corresponding structure maps from the category D(1)
This is a double-category analog of the operation of taking the “opposite” of a category:
another category in which morphisms are taken to be oriented in the opposite direction.
Indeed, together with such opposite operations in both horizontal and vertical directions,
the transpose generates a whole group of operations which take one double category to
another: it is plainly isomorphic to the dihedral group D4 , the symmetries of a generic
square, since each is determined by the source vertex of a generic square, together with
the sense of horizontal and vertical. The opposites do not illustrate much new structure,
however, so we will restrict our attention to the transpose.
The fact that the transpose is again an internal category in Cat is a standard consequence of basic facts about internalization. We do not want to assume all readers are
accustomed to such internal constructions, so will sketch a proof to convey the essential
idea.
e
4.7. Lemma. If D is a category internal in Cat, so is D.
Proof. First, one must check that the structure maps (e
e, se, e
t, e
◦) are functors. This follows
from the compatibility conditions for D. We give the example of the source functor se here:
the others follow similar lines.
e
We claim that the source functor in D
e (1) → D
e (0)
se : D
(51)
e (1)
preserves composition, that is, for two composable morphisms χ1 and χ2 in D
se(χ1 ◦ χ2 ) = se(χ1 ) ◦ se(χ2 )
(52)
e (1) and D
e (0) respectively. Interpreted in D, this is the
where the compositions are in D
statement that the composition functor ◦ : D(1) ×D(0) D(1) → D(1) is compatible with the
source maps in D(1) and D(0) , which is just part of the fact that ◦ is a functor. Likewise
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1455
se is compatible with the source, target, and identity inclusion maps in D(1) and D(0) .
Similar arguments prove the functoriality and compatibility of e
t, ee, and e
◦. So all the
e
structure maps (e
e, se, t, e
◦) are well-defined morphisms in Cat.
Furthermore, since the object and morphism maps of (e
e, se, e
t, e
◦) satisfy the axioms for
a category separately by definition, the functors satisfy them as well. Since its structure
e is indeed an internal
maps are well defined and satisfy all the axioms for a category, D
category in Cat.
In the above sketch, we deliberately chose to compare the interaction of different maps
in the two directions, namely source and composition, to emphasize the differences. It is
worth remarking that some conditions are symmetrical: for example, the fact that ◦ is
functorial, and therefore preserves composition, corresponds in the transpose to exactly
the same fact about e
◦. This is a form of the interchange law.
The above is a special case of a more general duality which can occur with internalization. Suppose we have any two types of objects, X and Y , which can be defined as
collections of objects and morphisms satisfying certain diagrammatic axioms. Then it
is equivalent to define X-objects internal to the category of Y -objects, and Y -objects
internal to the category of X-objects.
Next, we want use the transpose to better understand the structure of the transformation Cat-groupoid C//G associated to a 2-group action. We state the structure of the
]
transpose of C//G as a theorem. The essential new observation is that the transpose C/
/G
is built from transformation groupoids associated to group actions in the usual sense.
4.8. Theorem. Given a 2-group G, a category C, and an action of G on C given by
]
Φ̂ : G × C → C in Cat, the transpose of C//G is the category C/
/G internal to Gpd, with:
]
• Groupoid of Objects: (C/
/G)(0) , equal to the transformation groupoid C(0) //G (0) associated to the action given by Φ̂(0) .
]
• Groupoid of Morphisms: (C/
/G)(1) , equal to the transformation groupoid C(1) //G (1)
associated to the action given by Φ̂(1) .
• Identity inclusion functor
ee : C(0) //G (0) → C(1) //G (1)
(53)
se, e
t : C(1) //G (1) → C(0) //G (0)
(54)
e
◦ : (C(1) //G (1) ) ×C(0) //G (0) (C(1) //G (1) ) → C(1) //G (1)
(55)
source and target functors
and composition functor
are the functors whose object maps are the corresponding (e, s, t, ◦) for C and whose
morphism maps are those for G × C.
1456
JEFFREY C. MORTON AND ROGER PICKEN
Proof. Since this is a special case of the transpose construction of Lemma 4.7, we just
]
]
need to show the equalities for (C/
/G)(0) and (C/
/G)(1) . First, consider the action as
internal to Cat, by the inclusion of Gpd in Cat. Then the case of the groupoid of
objects is clear: its objects are the objects of x ∈ C, and its morphisms are objects of
(C//G)(1) = G × C, i.e. they are labeled by pairs (γ, x) ∈ G (0) × C(0) , and (γ, x), as a
]
morphism in (C/
/G)(0) , has source x and target γ I x. Composition is determined by
the object map of the composition functor in C//G (47) , i.e. by the monoidal product ⊗
of G (0) . But this is exactly the transformation groupoid C(0) //G (0) , where G (0) is a group
with product ⊗, acting on C(0) by Φ̂(0) .
]
The argument for the groupoid of morphisms (C/
/G)(1) is substantially the same. Its
objects are the morphisms of C, i.e. labelled by f ∈ C(1) , and its morphisms are pairs
((γ, χ), f ) with source f and target (γ, χ) I f . Composition in the groupoid of morphisms
is determined by the morphism part of the composition functor in C//G (47).
]
]
Since (C/
/G)(0) and (C/
/G)(1) are in fact groupoids, and all structure maps, being
]
functors, are also morphisms of Gpd, we have that C/
/G is a category internal to Gpd.
]
We would like to emphasize again, that in the transposed perspective of C/
/G, both
the category of objects and the category of morphisms are themselves transformation
groupoids in the usual sense, associated to group actions.
We are now ready to combine the perspectives given by C//G and its transpose to
express the whole local symmetry structure implied in the action of G on C in a balanced
way. This balanced viewpoint is expressed in our definition below of the transformation
double category C//G (abusing notation somewhat by keeping the same name as for the
internal category). We recall from subsection 2.7 the notation O for the set of objects, H
and V for the sets of horizontal and vertical morphisms, and S for the set of squares. By
the preceding discussion, the horizontal category is (C//G)(0) = C (Def. 4.3), the vertical
]
category is (C/
/G)(0) = C(0) //G (0) (Thm. 4.8), and for squares, the horizontal composition
of squares in S is the morphism map of ◦, while vertical composition of squares in S is
the morphism map of e
◦.
4.9. Definition. Given an action of a 2-group G on a category C, in terms of Φ or Φ̂
of definitions 3.2 and 3.3, the transformation double category C//G is given by:
• Objects: objects x ∈ O = C(0)
f
• Horizontal Category: (C//G)(0) = C, with morphisms x → y ∈ H = C(1)
composed as usual
(γ,x)
]
• Vertical Category: (C/
/G)(0) = C(0) //G (0) , with morphisms displayed as x −→
γ I x ∈ V = (C(0) //G (0) )(1) with source and target as shown and composed by
(γ 0 , γ I x) ◦ (γ, x) = (γ 0 γ, x)
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1457
]
• Squares: S = (C//G)(1) = (C/
/G)(1) = G (1) × C(1) , consisting of pairs of morphisms
(γ, χ), f , displayed as
from G and C, with elements of S denoted by
f
x
(γ,x)
/
(γ,χ),f
γIx
(γ,χ)If
(56)
y
(∂(χ)γ,y)
/ (∂(χ)γ)
Iy
with horizontal and vertical source and target as shown in the display. Horizontal
and vertical composition (equivalent to pasting of diagrams of the form (56)) is given
by:
(γ2 , χ2 ), g
(γ1 , χ1 ), f
(γ1 , χ2 χ1 ), g ◦ f
◦h
=
(57)
where γ2 = ∂(χ1 )γ1 , and:
(γ1 , χ1 ), (γ3 , χ2 ) I f
◦v
(γ3 , χ2 ), f
=
(γ1 γ3 , χ1 (γ1 B χ2 )), f
(58)
where the underlying squares in D(G) are given in (19).
4.10. Theorem. C//G is a well-defined double category.
Proof. This follows from general considerations, but can also be checked by using the
]
properties of the internal categories C//G and C/
/G, e.g. the vertical target of the vertical
composition of squares is well-defined due to (48). We note that the vertical target of the
horizontal composition of squares, equals the composition of the vertical targets of the
two squares, i.e.
(γ1 , χ2 χ1 ) I (g ◦ f ) = ((γ2 , χ2 ) I g) ◦ ((γ1 , χ1 ) I f )
(59)
by the functoriality of Φ̂ in (22). Likewise the horizontal target of the vertical composition
of squares equals the composition of the horizontal targets of the two squares, i.e.
(∂(χ1 )γ1 ∂(χ2 )γ3 , y) = (∂(χ1 (γ1 B χ2 ))γ1 γ3 , y).
(60)
This follows in a straightforward fashion from crossed module properties.
Associativity of the horizontal and vertical composition of squares is an immediate
]
consequence of the associativity of the composition functors in C//G and C/
/G. Finally
the interchange law for horizontal and vertical composition of squares corresponds to the
proof of functoriality of the composition functor in C//G (Theorem 4.4) and reduces to
stating the interchange law for four squares (49) in D(G).
1458
JEFFREY C. MORTON AND ROGER PICKEN
4.11. Remark. If we display the squares of the double category as follows:
x
(γ,x)
f
/
y
((γ,χ),f )
γIx
(γ,χ)If
(61)
/
(∂(χ)γ,y)
(∂(χ)γ) I y
with horizontal arrows dashed and vertical arrows dotted, this can be used to illuminate
the two internal category perspectives for the double category.
From the perspective of the internal category C//G of Definition 4.3, these squares of
the double category display the action of the source and target functors s, t on objects
(γ, x), (∂(χ)γ, y), and morphisms ((γ, χ), f ). The dashed arrows denote the image of the
square under source and target functors, while the dotted arrows denote its source and
target internal to the category of morphisms.
]
Similarly, from the perspective of the internal category C/
/G of Theorem 4.8, these
squares display the action of the source and target functors se, e
t on objects f , (γ, χ) I f ,
and morphisms ((γ, χ), f ). Now the dotted arrows denote the image of the square under
source and target functors, while the dashed arrows denote its source and target internal
to the category of morphisms.
The squares of the double category superimpose the two perspectives giving a balanced
view, which makes both internal structures transparent at the same time.
4.12. Structure of the Transformation Double Category. Next, we will consider some consequences of the construction of C//G. As constructed, the main information which can be read directly from C//G, viewed as an internal category in Cat, is
precisely the target map, which is just the action Φ̂ itself. Some less obvious consequences
]
are more easily read from the transpose C/
/G, or from C//G seen as a double category.
4.12.1. Relation to Ordinary Transformation Groupoids. We have seen, in
]
Theorem 4.8, that the groupoid of objects and the groupoid of morphisms of C/
/G are
]
both transformation groupoids. The structure maps of C//G relate these groupoids to each
other. Being functors, these maps consist of two parts, which we can consider separately.
Considering the identity-inclusion functor tells us how several transformation groupoids
are nested inside one another, and will justify the notation I we previously introduced
for our 2-group action. First, note what these groupoids are in crossed module notation.
]
4.13. Corollary. If G is given by a crossed module (G, H, ∂, B), then (C/
/G)(0) =
]
C(0) //G, and C(1) //G ⊂ (C/
/G)(1) = C(1) //(G n H).
Proof. In the construction of a categorical group from a crossed module, the group
G n H is the group of morphisms, and the group G of objects occurs as the subgroup of
elements of the form (γ, 1H ). The groupoid inclusion then follows immediately from the
identity-inclusion functor in Theorem 4.8.
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1459
The fact that the structure maps are functors means that, from an “external” point
of view (that is, if we look at underlying sets), C//G combines three closely related group
actions in the standard sense. These three group actions have all been denoted by the
symbol I introduced in Definition 3.5. This notation can now be justified because they
are all restrictions of the action of G (1) on C(1) , along the identity inclusion maps for G and
C. In particular, the associated action groupoids are also related by these restrictions.
More precisely, we have:
4.14. Corollary. The identity-inclusion functor ee factors into two inclusions:
]
]
(C/
/G)(0) ⊂ C(1) //G (0) ⊂ (C/
/G)(1)
(62)
where the first inclusion is given by the identity inclusion of C, and the second by the
identity inclusion of G.
Proof. The fact that objects of G give endofunctors of C means that the morphism
(1)
maps Φγ determine an action of G (0) on C(1) . Thus, objects of G act on both C(0) and
C(1) . By functoriality, the inclusion of objects of C as identity morphisms in C(1) is
compatible with the action. So the subset C(0) ⊂ C(1) is closed under the action of G.
The associated transformation groupoid has as its objects the morphisms f ∈ C, and as
morphisms pairs (γ, f ) which compose in the usual way. These correspond to the special
(γ, 1H ), f . There is a full inclusion of transformation groupoids induced by
squares
the inclusion of their sets of objects C(0) ⊂ C(1) .
The group G (1) ∼
= G n H also acts on C(1) . This action is given by Φ̂, and is related
to the action of G on C(1) by the identity inclusion map for G, namely G ⊂ G n H. In
particular, the action of G (1) on C(1) combines both natural transformations acting on
objects, and functors acting on morphisms. To see this, note that the squares in the
double category C//G occur by construction within a commuting cube which shows the
naturality square associated to f , or equivalently the two ways of expressing (γ, χ) I f
in terms of Φ - see (26):
f
x
/y
x
(63)
,/ y
f
(γ,x)
(γ,y)
((γ,χ),f )
(∂(χ)γ,x)
γIf
γIx
/
(∂(χ)γ,y)
γIy
Φ(γ,χ) (y)
Φ(γ,χ) (x)
&
∂(χ)γ I x
,
∂(χ)γIf
/
&
∂(χ)γ I y
1460
JEFFREY C. MORTON AND ROGER PICKEN
So the action of morphisms of G on morphisms of C involves natural transformations
at the source and target objects, as well as functors acting on the morphisms themselves.
The outside faces of the cube (63) are themselves special cases of squares in which one of
these two parts is an identity. The case χ = 1H , so that (γ, χ) = Idγ (for example, the
front or rear faces of that cube) is precisely the second inclusion of the theorem.
Restricting further to the image of the unit inclusion for C, we have the case f = Idx .
In particular, γ I Idx = IdγIx is just a consequence of the fact that the action is
functorial. (For this reason, it is immediate if we think of the action in terms of Φ̂.)
The action by the subgroup G gives a transformation groupoid with fewer morphisms
than the action by the big group G n H - but the former is strictly a sub-groupoid (by a
non-full inclusion) of the latter.
Notice that this factorization of ee is given by restricting to the image of the unit
inclusions of C and then G, and not the other way around. There is no well-defined
action of G (1) on C(0) . This can easily be seen, again by taking a special case of a square
of the form (63) with f = Idx . Since γ I x and ∂(χ)γ I x need not be the same
object, and even in the case they are, Φ(γ,χ) (x) need not be an identity, the image of
e : C(0) → C(1) is not fixed by the action of G (1) on C(1) .
4.14.1. Image of the Composition Functor. Next, consider the composition e
◦. It
is somewhat more complicated than ◦, and derives from the 2-group structure of G, just
as composition does in transformation groupoids. A little example calculation with this
composite will show how 2-group structure and an action must cohere.
]
Indeed, composition in (C/
/G)(0) is exactly the composition in the transformation
]
groupoid C(0) //G, by Corollary 4.13. Similarly, the composition in (C/
/G)(1) is the com(1)
position in C //G (1) , which is determined by the group multiplication in G (1) ∼
= GnH
(in the crossed-module representation), together with the action on C(1) . So we can write
this as:
(γ 0 , χ0 ), (γ, χ) I f
e
◦
(γ, χ), f
=
(γ 0 γ, χ0 (γ 0 B χ)), f
(64)
We know by construction and the above theorems that this makes a well-defined double
category. It is, however, not given directly by the action Φ seen as a 2-functor. In that
point of view, the action of morphisms of G on morphisms of C is a consequence of the
naturality of Φ(γ,χ) for all (γ, χ) ∈ G (1) .
Writing the target of this composite concretely, it is:
(γ 0 γ, χ0 (γ 0 B χ)) I f
(65)
However, we have concrete expressions for (γ, χ) I f , shown in the cube (63), namely:
(γ, χ) I f = (∂(χ)γ I f ) ◦ Φ(γ,χ) (x)
=
Φ(γ,χ) (y)
◦ (γ I f )
(66)
where composition is taken in C. Recall that this is a consequence of the naturality of
the bottom square of (63) when we apply Φ(γ,χ) at the morphism f .
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1461
Now, if we then take (γ, χ) I f as the source of the new square (γ 0 , χ0 ), (γ, χ) I f
to find its target, we are applying Φ(γ 0 ,χ0 ) at the morphism (γ, χ) I f . To compute this
explicitly, note that each morphism in the commuting square at the bottom of (63) gives
rise to another commuting square, and these form four faces of a cube whose other two
faces are the image of the original commuting square under the functors Φγ 0 and Φ∂χ0 γ 0
respectively:
∂(χ0 )γ 0 γIf
∂(χ0 )γ 0 γ Ii x
/
∂(χ0 γ 0 γ) Ii y
Φ(γ 0 ,χ0 ) (γIy)
Φ(γ 0 ,χ0 ) (γIx)
γ 0γ I x
/ γ 0γ
∂(χ0 )γ 0 IΦ(γ,χ) (x)
γ 0 γIf
Iy
∂(χ0 )γ 0 IΦ(γ,χ) (y)
γ 0 IΦ(γ,χ) (x)
∂(χ0 )γ 0 ∂(χ)γIf
∂(χ0 )γ 0 ∂(χ)γ i I x
/
$
γ 0 IΦ(γ,χ) (y)
∂(χ0 )γ 0 ∂(χ)γ i I y
Φ(γ 0 ,χ0 ) (∂(χ)γIy)
Φ(γ 0 ,χ0 ) (∂(χ)γIx)
γ 0 ∂(χ)γ I x
γ 0 ∂(χ)γIf
/ γ 0 ∂(χ)γ
Iy
(67)
We have omitted the images under Φγ 0 and Φ(∂χ0 )γ 0 of the original diagonal (γ, χ) I f for
clarity. However, they are the diagonals on the front and back face of this cube. Together
with the corresponding edges between these faces, they form a square whose diagonal is
the target we want, just as before.
(γ 0 , χ0 ), (γ, χ) I f , hence also of our comThe target morphism of the square
posite square, is therefore the dotted arrow across the diagonal of this cube.
The source object of this morphism is γ 0 γ I x, and its target object is ∂(χ0 )γ 0 ∂(χ)γ I
y.
(γ 0 γ, χ0 (γ 0 B χ)), f .
However, we already found that the composite square was
Computing the target of this square directly from (63) shows that its target morphism
begins at γ 0 γ I x and ends at (∂(χ0 (γ 0 B χ))γ 0 γ I x. A straightforward application of
the crossed module axioms confirms that indeed:
∂(χ0 )γ 0 ∂(χ)γ = ∂(χ0 (γ 0 B χ))γ 0 γ
(68)
as it must be.
Knowing that we are in a double category where composition is well-defined tells us
more. Just as the naturality square at the bottom of (63) gave us two expressions for
(γ, χ) I f , the cube of naturality squares (67) gives us a total of six expressions which
equal the morphism
(γ 0 γ, ∂(χ0 (γ 0 B χ))) I f
(69)
1462
JEFFREY C. MORTON AND ROGER PICKEN
As before, each is a composite of morphisms in C, most directly written as:
=
=
=
=
=
Φ(γ 0 ,χ0 ) (∂(χ)γ I y)
(∂(χ0 )γ 0 ) I Φ(γ,χ) (y))
Φ(γ 0 ,χ0 ) (∂(χ)γ I y)
(∂(χ0 )γ 0 ∂(χ)γ I f )
(∂(χ0 )γ 0 I Φ(γ,χ) (y))
(∂(χ0 )γ 0 ∂(χ)γ I f )
◦
(γ 0 I Φ(γ,χ) (y))
◦
Φ(γ 0 ,χ0 ) (γ I y)
◦
(γ 0 ∂(χ)γ I f )
◦ Φ(γ 0 ,χ0 ) (∂(χ)γ I x)
◦
(∂(χ0 )γ 0 γ I f )
◦ (∂(χ0 )γ 0 I Φ(γ,χ) (x))
◦
(γ 0 γ I f )
◦
(γ 0 γ I f )
0
◦ (γ I Φ(γ,χ) (x))
◦ (γ 0 I Φ(γ,χ) (x))
◦ Φ(γ 0 ,χ0 ) (γ I x)
◦ Φ(γ 0 ,χ0 ) (γ I x)
(70)
The fact that these are all equal is a consequence of the fact that Φ is an action, hence
every square in the cube is either a naturality square or the image of a naturality square
under a functor, and therefore every square commutes. Some of the terms may also be
written differently in light of (68) and similar applications of crossed-module axioms.
In general, a composite of n squares will give rise to many different equivalent expressions for the target morphism, in the same way. In particular, one can easily see
by induction that one obtains an (n + 1) cube by repeatedly applying Φ at each step.
Counting paths around such a cube, one finds (n + 1)! possible composites in C which
amount to the same morphism.
Clearly, using the 2-group composition as in (64) to find the target (65) is more direct,
and since we know the transpose of C//G is a double category, we can use the 2-group
action to find the target directly.
4.14.2. Horizontal and Vertical 2-Categories. Finally, we turn to the double
category point of view on C//G. When speaking of double categories, one often refers
to various smaller structures which arise from specialization. We already know that the
horizontal and vertical categories are, respectively, C itself, and C(0) //G (0) . Moreover,
there are also two categories which share the same collection of morphisms (the squares
of the double category), but have different sets of objects (namely, the horizontal and
vertical morphisms). Again, we have already seen that these are, respectively, G × C and
C(1) //G (1) .
However, one also speaks of the horizontal and vertical 2-categories of a double category. These are the last aspect of the structure of C//G we will consider.
4.15. Definition. The horizontal 2-category H(D) of a double category D has the same
objects and 1-morphisms as its horizontal category. The 2-morphisms consist only of the
squares of D for which the vertical morphisms on the boundary are identities. The vertical
e
2-category V (D) is defined similarly (i.e. it is the horizontal 2-category of D).
It is straightforward to check that these form 2-categories, but since it is a standard
fact, we will not verify all the details here. However, we will consider the horizontal and
vertical 2-categories for C//G. It should be clear from what we have just said that these
will amount to C and C(0) //G (0) , extended by adding certain 2-morphisms.
4.16. Theorem. The horizontal 2-category H(C//G) of the transformation double category associated to a 2-group action has the same objects and 1-morphisms as C. Given
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1463
objects x, y ∈ C, and a pair of morphisms f, f 0 : x → y, the 2-morphisms in Hom(f, f 0 )
are labeled by (1G , χ) : 1G ⇒ 1G such that
f 0 = f ◦ Φ(1G ,χ) (x)
(71)
These 2-morphisms compose horizontally and vertically just as the endomorphisms of 1G
in G.
Proof. The objects and 1-morphisms are those of the horizontal category, which is just
C by definition.
The 2-morphisms correspond to squares of the form (56) for which the vertical mor(1G , χ), f , since the horizontal
phisms are identities. These must be of the form
(γ, χ), f
source of
is (γ, x) = Idx , and thus γ = 1G . Moreover, since the target
(∂(χ)γ, y) = (∂(χ), y) = Idy , we must also have that ∂(χ) = 1G . So only 2-endomorphisms
of the identity of G enter as 2-morphisms.
The vertical target is then the horizontal morphism
(1G , χ) I f = (∂(χ) I f ) ◦ Φ(1G ,χ) (x)
= f ◦ Φ(1G ,χ) (x)
(72)
When γ1 and γ2 in (57) and (58) are both equal to 1G , these horizontal and vertical
composition rules just amount to multiplication in H, that is, the composition of 2endomorphisms of the identity.
Intuitively, this says that the 2-morphisms with source f in H(C//G) correspond exactly to the endomorphisms (1G , χ) of 1G in G. In the language of crossed modules, they
correspond to χ ∈ ker(∂). The target of such a 2-morphism is found by precomposing f
with the endomorphisms of x given by the natural transformations Φ(1G ,χ) . Clearly, these
labels χ for 2-morphisms sourced at f will be the same for all f . Depending on the precise
details of the action Φ, all such 2-morphisms might be endomorphisms of f itself, or the
targets might all be different from f and from each other. This is the type of information
about the action Φ which is captured by H(C//G).
4.17. Theorem. The vertical 2-category V (C//G) of the transformation double category
associated to a 2-group action has the same objects and 1-morphisms as C(0) //G (0) . Given
an object x ∈ C and γ, γ 0 ∈ G such that γ I x = γ 0 I x, there will be a 2-morphism from
(γ, x) to (γ 0 , x) for each χ such that Φ(γ,χ) (x) = IdγIx and γ 0 = ∂(χ)γ.
Proof. The objects and 1-morphisms are those of the vertical category, which is C(0) //G (0)
by Theorem 4.8.
The 2-morphisms correspond to squares of the form (56) in which x = y and the
horizontal morphisms on the boundary are both identities. Since the vertical source is
(γ, χ), Idx .
f = Idx , they are squares of the form
Moreover, the vertical target must also be the identity. Now, to begin with, this must
mean that γ I x = γ 0 I x (taking γ 0 = ∂(χ)γ), as stated in the theorem. This vertical
1464
JEFFREY C. MORTON AND ROGER PICKEN
target is the horizontal morphism
(γ, χ) I f = (γ 0 I Idx ) ◦ Φ(γ,χ) (x)
= Idγ 0 Ix ◦ Φ(γ,χ) (x)
(73)
The second equality holds because γ 0 acts on f = Idx by the functor Φγ 0 , and functors
respect identities. Thus, this condition implies that not only are γ I x and γ 0 I x equal,
but indeed the natural transformation Φ(γ,χ) intertwines them by the identity on γ I x.
The compositions again follow (57) and (58).
Intuitively, the transformation groupoid C(0) //G (0) displays the object part of the action Φ̂. The morphism part can potentially appear here also, relating local symmetry
transformations which carry x to the same place. However, as we have seen, such a relation need not exist if γ 6= γ 0 , since even though γ I x = γ 0 I x, there might be no natural
transformation intertwining them, or none intertwining them with identities. Conversely,
there may be more than one such natural transformation. This is the information about
the full action Φ captured by V (C//G) which the mere groupoid C(0) //G (0) cannot see.
We would next like to explicitly describe the above, by examining the structure of the
transformation double groupoid G//G, associated to our example of the adjoint action of
a 2-group G on itself.
4.18. The Transformation Double Category for the Adjoint Action. As in
all transformation double groupoids, the horizontal category of G//G is just the same as G,
seen as a category (which happens to be monoidal). The vertical morphisms and squares
will always be labelled by pairs of, respectively, objects and morphisms from G.
Summing up the above, we have:
4.19. Corollary. The transformation double groupoid G//G for the adjoint action of a
2-group G, classified by the crossed module (G, H, B, ∂), has:
• Objects: labelled by g ∈ G
• Horizontal Category: the underlying category of G
• Vertical Category: same as G//G, the transformation groupoid for the adjoint action
of G
• Squares: labelled by pairs ((γ1 , χ1 ), (g1 , η1 )) ∈ (G × H) × (G × H), denoted by
(γ1 , χ1 ), (g1 , η1 ) . As in (56), the horizontal and vertical source and target of
this square are displayed as follows:
g1
Φγ1
η1
(γ1 ,χ1 ),(g1 ,η1 )
g3
η2
/ g2
Φγ2
/ g4
(74)
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1465
where γ2 = ∂(χ1 )γ1 . Here η2 is determined by:
g3
η2
=
g4
γ1
g1
γ1−1
χ1
η1
.
χ−h
1
γ2
g2
γ2−1
(75)
• Horizontal composition of squares is given by multiplication in H:
(γ2 , χ2 ), (g2 , η2 )
◦h
(γ1 , χ1 ), (g1 , η1 )
=
(γ1 , χ2 χ1 ), (g1 , η2 η1 )
(76)
where g2 = ∂(η1 )g1 , and vertical composition of squares is given by:
(γ1 , χ1 ), (γ3 , χ2 ) I (g1 , η2 )
◦v
(γ3 , χ2 ), (g1 , η2 )
=
(γ1 γ3 , χ1 (γ1 B χ2 )), (g1 , η2 )
(77)
which makes them into the transformation groupoid (G n H)//(G n H),
Proof. This is a direct consequence of the structure of C//G laid out in Definition 4.9
and Theorem 4.10, with the structure of G as a categorical group given explicitly for C.
Now we will find the horizontal and vertical 2-categories, H(G//G) and V (G//G), of
the double category associated to the adjoint action of G
4.20. Corollary. If G is a 2-group classified by the crossed module (G, H, B, ∂), then
the horizontal 2-category H(G//G) associated to the adjoint action of G has:
• Objects: g ∈ G
• Morphisms: (g, η) ∈ G n H with source, target, and composition as in G
• 2-Morphisms: ((g, η), χ) ∈ (G n H) × ker(∂), whose source is (g, η), and whose
target is
(g, η 0 ) = (g, η · (χg B χ−1 ))
(78)
Composition is given by multiplication in the subgroup ker(∂) ⊂ H.
Proof. Taking the expression (71) for the target of a 2-morphism, in the case of the
adjoint action we use the fact that Φ(1G ,χ) (g) is defined in (38). Substituting this gives
the target specified in (78). Composition as multiplication in H also follows from Theorem
4.16.
1466
JEFFREY C. MORTON AND ROGER PICKEN
This tells us how to extend G to a 2-category to reflect part of the symmetry that
arises from its action on itself. The vertical 2-category, by contrast, can be better seen
as an extension of G//G, the transformation groupoid for the actions of just the group of
objects on itself. In particular, we have the following.
4.21. Corollary. If G is a 2-group classified by the crossed module (G, H, B, ∂), then
the vertical 2-category V (G//G) associated to the adjoint action of G has:
• Objects: g ∈ G
• Morphisms: (γ, g) ∈ G × G with source, target, and composition maps as in the
transformation groupoid for the adjoint action of G on itself
• 2-Morphisms: There will be a 2-morphism from (γ, g) to (γ 0 , g), for each χ ∈ H
such that
(γgγ −1 ) B χ−1 = χ−1
(79)
(That is, χ−1 is a fixed point for γgγ −1 under the action B).
Proof. In Theorem 4.17, we established when there is a 2-morphism in V (C//G). In our
case, taking x = g, we find there is such a 2-morphism for any χ such that Φ(γ,χ) (g) =
Idγgγ −1 .
But we have the expression (38) for the maps defining the natural transformation
Φ(γ,χ) . So this condition says that
χ(γgγ −1 ) B χ−1 = 1H
(80)
(γgγ −1 ) B χ−1 = χ−1
(81)
or in other words, that
This is exactly the statement above.
As usual, composition follows (57) and (58).
Intuitively, this tells us that the information about the action Φ captured by the
2-morphisms relating two different symmetry relations γ and γ 0 taking g to g is fundamentally information about the crossed module structure, and the action B of G on
H.
We have included these calculations of the horizontal and vertical 2-categories, partly
because these particular slices of the double category G//G (or C//G generally) are interesting in their own right. But we also include them to illustrate that many squares in the
double category do not appear in either the horizontal or vertical 2-categories. Each is
missing a great deal of information about the action Φ, and we need to go to the double
category G//G (or C//G in general) for the full information.
TRANSFORMATION DOUBLE CATEGORIES ASSOCIATED TO 2-GROUP ACTIONS
1467
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Department of Mathematics and Statistics,
The University of Toledo, 2801 W. Bancroft St.
Toledo, OH 43606-3390, USA
Center for Mathematical Analysis, Geometry and Dynamical Systems,
Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: jeffrey.c.morton@theoreticalatlas.net
roger.picken@tecnico.ulisboa.pt
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